How do I animate a single face of a mesh?
You cannot keyframe vertices directly, but several possibilities to do that.

For instance:


Using an armature: 


Create an armature, create bones in it and assign the wanted names to the bones (ie states names), in the map assign the wanted vertices to groups which are named the same way. Parent the map to the armature using 'with empty groups' option (as the groups are already created).



pros/cons: can grab, rotate, scale the parented group as you wish


Using a shrinkwrap:


Add a shrinkwrap modifier which applies on a vertex group. Here you can animate the plane target use in the shrinkwrap.

pros/cons: fast to do but static: only up and down movements




Using shapekeys:


Add a base shapekey, then a second one and this last selected, move the vertices to the target place you want. Now you can animate the shape key.



pros/cons: you can place the vertices exactly where you need, but static once done


Using scale:


Here, simply separate the country or state you want and play with its scale.

pros/cons: only for up/down movements





Edit:


Using a hook:


In edit mode, select the vertices you want and CtrlH to hook them:



pros/cons: nearly like bone here: grab, rotate, scale the hooked vertices (or group) as you wish


Using a lattice


Again with a vertex group but here with the lattice modifier.



pros/cons: not the simplest to setup but can grab and scale too.